Calculate all the primitive roots of 41 and 26.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Educational Material: Number Theory and Primitive Roots
#### Problems and Exercises
**9.** Prove that if \( n \) is a perfect square, then \( \lambda(n) = 1 \).
**10.** Let \( r \) be a primitive root of some \( n \geq 3 \). Prove that:
\[
r^{\frac{\phi(n)}{2}} \equiv \pm 1 \pmod{n}
\]
**11.** Find \( \tau(n), \sigma(n), \lambda(n), \mu(n), \omega(n), \) and \( \phi(n) \) for the following integers:
- 2250
- 199
- 286936650
- 22!
**12.** Let \( p = 17 \) and \( d \) be a divisor of \( \phi(p) \). Determine \( \psi(d) \) for each \( d \). List all divisors, \( d \), of \( \phi(p) \).
**13.** Calculate all the primitive roots of 41 and 26.
**14.** Demonstrate that 21 has no primitive root.
- Notes show attempts considering primes such as 2, 3, 5, 7, 11, 21.
**15.** Let \( r \) be a primitive root of \( n \). If \(\gcd(a, n) = 1\), then the smallest power \( k \) such that \( r^k \equiv a \pmod{n} \) is called the **index of \( a \) relative to \( r \)**, denoted by \(\text{ind}_r(a)\). This concept is useful to solve congruences. Consider the properties of indices.
- Solve: \( 8x^4 \equiv 11 \pmod{13} \).
#### Annotations and Remarks
- The problems are sequentially numbered with some highlighted or circled for emphasis.
- There are handwritten notes alongside problems indicating checks, attempted solutions, or additional thoughts like "150, 157 maybe theorem".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F8eb1a4a4-350c-4084-86a7-e66dd7cb95fb%2Fiv9schq.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Material: Number Theory and Primitive Roots
#### Problems and Exercises
**9.** Prove that if \( n \) is a perfect square, then \( \lambda(n) = 1 \).
**10.** Let \( r \) be a primitive root of some \( n \geq 3 \). Prove that:
\[
r^{\frac{\phi(n)}{2}} \equiv \pm 1 \pmod{n}
\]
**11.** Find \( \tau(n), \sigma(n), \lambda(n), \mu(n), \omega(n), \) and \( \phi(n) \) for the following integers:
- 2250
- 199
- 286936650
- 22!
**12.** Let \( p = 17 \) and \( d \) be a divisor of \( \phi(p) \). Determine \( \psi(d) \) for each \( d \). List all divisors, \( d \), of \( \phi(p) \).
**13.** Calculate all the primitive roots of 41 and 26.
**14.** Demonstrate that 21 has no primitive root.
- Notes show attempts considering primes such as 2, 3, 5, 7, 11, 21.
**15.** Let \( r \) be a primitive root of \( n \). If \(\gcd(a, n) = 1\), then the smallest power \( k \) such that \( r^k \equiv a \pmod{n} \) is called the **index of \( a \) relative to \( r \)**, denoted by \(\text{ind}_r(a)\). This concept is useful to solve congruences. Consider the properties of indices.
- Solve: \( 8x^4 \equiv 11 \pmod{13} \).
#### Annotations and Remarks
- The problems are sequentially numbered with some highlighted or circled for emphasis.
- There are handwritten notes alongside problems indicating checks, attempted solutions, or additional thoughts like "150, 157 maybe theorem".
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